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 A320744 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 4 or fewer colors (subsets). 3
 0, 0, 0, 0, 0, 6, 30, 130, 532, 2006, 7626, 28401, 106260, 396435, 1486147, 5580130, 21032880, 79486763, 301317282, 1145123672, 4362804633, 16658456825, 63738451998, 244332656201, 938244497740, 3608640426930, 13899977105315, 53614228550220, 207061964668740, 800639722002163, 3099251007215286, 12009598156277090, 46582685655751645, 180850428684482360 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Two color patterns are equivalent if the colors are permuted. Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference. There are nonrecursive formulas, generating functions, and computer programs for A056292 and A305750, which can be used in conjunction with the first formula. LINKS Andrew Howroyd, Table of n, a(n) for n = 1..200 E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665. FORMULA a(n) = (A056292(n) - A305750(n)) / 2 = A056292(n) - A056354(n) = A056354(n) - A305750(n). a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=4 is the maximum number of colors, Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)), and A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)). a(n) = A059053(n) + A320643(n) + A320644(n). EXAMPLE For a(6)=6, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, AABACC-AABBAC, AABACD-AABCAD, and AABCBD-AABCDC. MATHEMATICA Adnk[d_, n_, k_] := Adnk[d, n, k] = If[n>0 && k>0, Adnk[d, n-1, k]k + DivisorSum[d, Adnk[d, n-1, k-#]&], Boole[n == 0 && k == 0]] Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]] (* A304972 *) k=4; Table[Sum[(DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j]&]/n - Ach[n, j])/2, {j, k}], {n, 40}] CROSSREFS Column 4 of A320742. Cf. A056292 (oriented), A056354 (unoriented), A305750 (achiral). Sequence in context: A007465 A261389 A073389 * A232061 A247386 A317755 Adjacent sequences: A320741 A320742 A320743 * A320745 A320746 A320747 KEYWORD nonn,easy AUTHOR Robert A. Russell, Oct 21 2018 STATUS approved

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Last modified March 28 07:43 EDT 2023. Contains 361577 sequences. (Running on oeis4.)